A large noncollinear magnetoresistance (NCMR) is observed for Rh/Co atomic bilayers on Ir(111) using scanning tunneling microscopy and spectroscopy. The effect is 20\% at the Fermi energy and large in a broad energy range. The NCMR can be used to electrically detect nanometer-scale domain walls and skyrmions directly in the tunnel current without the need for a differential measurement. The NCMR results from changes in the density of states of noncollinear spin textures with respect to the ferromagnetic state. Density functional theory calculations reveal that they originate from spin mixing between majority d(xz) and minority p(z) states.